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Calamities and Kenneth Feinberg have a way of finding each other.As soon as the Deepwater Horizon drilling rig sank and triggered a huge oil spill in the Gulf of Mexico in April, Mr. Feinberg started mulling how an avalanche of claims by economic victims could be handled. Less than two weeks ago, BP PLC called to offer him a job doing just that. Last week, BP and President Barack Obama jointly chose him as administrator of an unprecedented $20 billion compensation fund.

The Washington lawyer and arbitration guru is now racing to speed up the system that BP initially put in place to compensate tens of thousands of Gulf residents and businesses for their financial injuries. It's a task he predicts will outstrip in complexity all the other victims' funds he has run, from the Agent Orange liability litigation in the 1980s to the $7 billion for victims of the September 2001 terror attacks.

What makes this case all the harder to handle is that it remains an unfolding disaster. "Until the oil stops, this is an ongoing tort. We just can't say what the final size of claims will be," Mr. Feinberg said in an interview. "The sheer scope of this is something we haven't seen before."

Two days after BP and the White House agreed on June 16 to set up an independent claims fund, Mr. Feinberg flew to the region, first to Mississippi and then Louisiana. Mr. Feinberg met Monday with BP executives in Houston to work out details for taking over the claims process. He is set to travel next to Alabama and Florida.

The fund he is set to administer has a number of oddities. It was created as a voluntary compact between the U.S. government and BP, but without any act of Congress, executive order or other legal anchor. Mr. Feinberg will have complete latitude to determine the fund's operations, including all key decisions as to who gets money and in what amounts.

Since his appointment, Mr. Feinberg has said repeatedly that he would neither take orders from the White House nor BP. But he said that Mr. Obama did give him one instruction: to get payments out quickly.

After meeting with Mississippi Gov. Haley Barbour on Friday, the 64-year-old Mr. Feinberg warned of the high potential for fraudulent claims in the same breath as he urged anyone who thinks they may be eligible to step forward. Getting people to apply in the first place is his first big hurdle.

"You'd be amazed how many people don't come forward," he said. "They procrastinate, they worry, they doubt they will qualify, they don't trust the process, whatever."

BP has faced heavy criticism for lapses leading up to the disaster and for failing to prepare for a spill of this size. But Mr. Feinberg praised the company for its work in getting a claims process running swiftly. For at least another month, Mr. Feinberg will be advising BP on ways to speed up payments and make the process more transparent. He said he hoped to assume all responsibilities by August.

In seven weeks, BP has written 31,000 checks worth $104 million to Gulf Coast fishermen, shrimpers and others whose work has been cut short by the spill. A team of nearly 1,000 clerks and adjusters are processing claims in 33 field offices from Louisiana to Florida.

"BP deserves a fair amount of credit here," Mr. Feinberg says. "This is the first time I know of where a company has implemented a whole process like this in the midst of an ongoing crisis."

The son of a tire salesman and a bookkeeper from Brockton, Mass., Mr. Feinberg makes his case in the heavy twang of south Boston. A former chief of staff to the late Sen. Ted Kennedy, he went on to become one of the country's most prominent arbitrators. After a rough start, he earned wide respect for how he handled the survivors and families of those killed on 9/11. The Treasury Department appointed him last year to monitor executive pay at companies that had received federal bailout money during the financial crisis.

"Ken is accustomed to handling these sort of issues without emotion," said former Attorney General John Ashcroft, who appointed Mr. Feinberg to run the 9/11 fund. "He was very good at distinguishing the deserving from the undeserving."

Doing the same along the Gulf Coast could be his biggest challenge. Can a motel 100 miles from the coast file for diminished revenues? What about a restaurant in Atlanta now having to pay more for shrimp?

And what about a New Orleans strip bar called the Mimosa Dancing Girls, which has put in a claim for a slump in sales? Asked about that one Monday on "Good Morning America," Mr. Feinberg said he was "very dubious about that claim," but didn't want to prejudge it.

As a rule, Mr. Feinberg said, he will turn to state law for guidance on which types of claims to honor and which to dismiss.

In the end, one aim of the fund—and a prime reason BP agreed to it—will be to minimize lawsuits against the company. To do that, Mr. Feinberg will offer big lump-sum payments to workers and businesses as an enticement to stay out of court.

"At some point, I will have to make an offer—'You take this amount in full satisfaction of your claim, but only if you waive your right to future litigation,'" Mr. Feinberg said. "And if I package it right, people will see that it makes no sense to fight it out in court."